Homework problems
More sufficiency of sequences:
A topological space X is sequentially compact if every sequence in X
has a convergent subsequence.
Please read and think a little bit about the following assertion:
- a compact metric space is sequentially compact
- a sequentially compact metric space is compact
NB: The take-away is that in metric spaces, sequential compactness
and compactness are equivalent.
NB: While this is treated in detail in section 28 of Munkres, I
believe a more streamlined argument is possible than the one given
there.
Separation in metric spaces:
Recall that a topological
space is regular if a point and a closed set can be separated by open
sets; a topological space is normal if two closed sets can be
separated by open sets. See section 31 of Munkres for precise
definitions.
We observed the elementary result that a metric space is Hausdorff.
We also have the following stronger result, which you find as Theorem
32.2 in Munkres:
Indeed, it is possible to prove this result quite simply by proving a
version of the Urysohn lemma for metric spaces. See p212.3 of Munkres
for additional details. A relevant fact concerning metric spaces is
the following result. For a metric space X and a subset A of X define
d(x,A)=infimum d(x,a), where the infimum is taken over all points a in
A. We have:
- d(x,A) is a continuous function of x
- d(x,A)=0 if and only if x is in the closure of A
- use this to understand p212.3 in Munkres